Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Measure how conditioning changes likelihood
Explain conditional probability and independence in everyday language and situations.
Without replacement, the first outcome changes both favorable and total counts for the second draw. We’ll compute the original second-red probability, update the bag after a first red, compute the …
Preview problemDistinguish a conditional probability from its reverse
Explain conditional probability and independence in everyday language and situations.
Reversed conditionals share the same overlap numerator but use different conditioned totals. We’ll compute the test-positive rate within the condition row, reverse the direction and use the positive-test column, reduce …
Preview problemCompute conditional probability from a finite sample space by restricting to the given event
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
Conditioning replaces the original sample space with only outcomes satisfying the given event. We’ll restrict the list first, count the retained outcomes for the denominator, count target outcomes inside that …
Preview problemCompute a conditional probability from Venn-diagram counts
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
In a Venn diagram, conditioning on B makes the whole B circle the denominator. We’ll total its overlap and B-only regions, use the overlap as the favorable A count, reduce …
Preview problemFind a conditional probability from a two-way table by restricting to the given event
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
A table conditional uses the margin of the named condition, not the target column or grand total. We’ll restrict to the freshman row, place its bus cell over the row …
Preview problemFind a conditional probability from the overlap and the given event
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
Conditional probability is the overlap probability divided by the condition probability. We’ll place those percentages in the correct roles, cancel their common percent units, reduce the numerical ratio, convert it …
Preview problemFind a joint count from a conditional probability and a given condition count
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
A conditional rate times its conditioned-group count recovers the joint count. We’ll rearrange the favorable-over-group relationship, multiply the rate by the B total, check that the result is a whole …
Preview problemFind the given-event total from a joint count and a conditional probability
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
When the joint count and conditional rate are known, the conditioned-group total is the unknown denominator. We’ll write the rate equation, clear that denominator, divide the known joint count by …
Preview problemEvaluate and compare two conditional rates
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
Comparable conditional rates measure the same outcome within different groups. We’ll align the studied and not-studied pass rates, identify which is larger, subtract smaller from larger, and express the absolute …
Preview problemIdentify the numerator and reference subgroup in a conditional probability
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
A conditional fraction names its reference subgroup in the denominator and its target-within-group count in the numerator. We’ll identify seniors as the condition, art-liking seniors as the favorable subset, state …
Preview problemCompute conditional probability by restricting to the given event
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
The phrase among sports participants establishes the entire conditioned group before any fraction is formed. We’ll identify soccer-playing sports participants as the numerator subset, use all sports participants as the …
Preview problemDecide which option is better supported by data using conditional probability
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
Because both conditionals measure recovery within their respective groups, their rates are directly comparable. We’ll convert and align the two percentages, identify the higher observed rate, compute the percentage-point gap, …
Preview problemDecide whether a conditional probability is possible, impossible, or undefined
Compute conditional probability as the fraction of one event's outcomes that also belong to another event.
A defined conditional probability is a part-to-whole fraction within the conditioned group, so it must lie from zero through one. We’ll test the proposed value against that range, interpret it …
Preview problemUse the Addition Rule to find P(A or B) for overlapping events
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Adding two event probabilities counts their shared outcomes twice, so the addition rule subtracts the overlap once. We’ll substitute both marginals and the intersection, perform the correction, and check that …
Preview problemFind \(P(A or B)\) for mutually exclusive events
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Mutually exclusive events have zero overlap, so the general addition rule simplifies cleanly. We’ll set the intersection term to zero, add the two disjoint probabilities, and express the resulting inclusive …
Preview problemFind P(A and B) from P(A), P(B), and P(A or B) using the Addition Rule
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
The overlap can be recovered by rearranging the addition rule before inserting numbers. We’ll isolate the intersection as the sum of marginals minus the union, substitute in that order, and …
Preview problemFind a missing event probability using the Addition Rule
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Solving the addition rule for one marginal changes the overlap’s sign when it crosses the equation. We’ll isolate the missing event probability symbolically, substitute union, other marginal, and intersection, evaluate …
Preview problemFind P(A or B) from Venn-diagram information using the Addition Rule
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
A Venn union is the three disjoint regions inside at least one circle. We’ll add A-only, overlap, and B-only once each, exclude the outside region from the numerator while using …
Preview problemFind P(A or B) from totals that include an overlap
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Table margins for A and B each include the shared cell, so adding them double-counts that overlap. We’ll subtract the joint count once, evaluate the corrected union count, divide by …
Preview problemInterpret an inclusive union probability in context
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Probability uses inclusive or unless exclusivity is stated, so the overlap remains included. We’ll translate the union into the three contextual groups, convert the decimal to a percent, and phrase …
Preview problemIdentify whether “A or B” means inclusive or exclusive in a probability statement
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Standard probability or is inclusive and therefore covers three regions: A-only, B-only, and the overlap. We’ll identify those regions, keep shared outcomes once, and separate the meaning of union from …
Preview problemFind the probability that neither event occurs using the Addition Rule and the complement
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Neither event means the complement of their inclusive union. We’ll translate the wording into that complement, subtract the known union probability from one, convert the remainder to a percent, and …
Preview problemDecide whether given probabilities for two events are possible using the Addition Rule
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
Every probability, including a union probability, must lie between zero and one. We’ll test the proposed union against that universal bound, interpret it as a percent to expose the contradiction, …
Preview problemChoose the correct Addition Rule setup for A or B and neither probability questions
Apply and interpret the Addition Rule P(A or B)=P(A)+P(B)-P(A and B).
The correct union setup must repair the double-counted overlap created by adding both marginals. We’ll begin with that sum, subtract one copy of the intersection, and distinguish this addition rule …
Preview problemFind a joint probability from a marginal probability and a conditional probability
Apply and interpret the general Multiplication Rule with conditional probability.
A joint path equals the first event’s probability times the second event’s conditional probability after the first. We’ll write the multiplication rule in that order, substitute and multiply, and divide …
Preview problemFind a joint probability by multiplying a marginal probability and a conditional probability
Apply and interpret the general Multiplication Rule with conditional probability.
The multiplication rule can begin with B and then use A given B along the same path. We’ll preserve that marginal-then-conditional order, multiply the two rates for the joint event, …
Preview problemCompute a without-replacement two-step probability using the Multiplication Rule
Apply and interpret the general Multiplication Rule with conditional probability.
Without replacement, the second branch must use the deck remaining after the first favorable draw. We’ll form the first-ace probability, decrement both ace and card counts, multiply the updated conditional …
Preview problemFind the probability that two independent events both occur
Apply and interpret the general Multiplication Rule with conditional probability.
Both specified outcomes from independent devices form an intersection whose probability is the product of their marginals. We’ll record the coin and die probabilities, multiply the fractions, and verify the …
Preview problemFind a conditional probability from a joint probability and a marginal probability
Apply and interpret the general Multiplication Rule with conditional probability.
To find B given A, divide the joint probability by A’s marginal because A supplies the conditioned group. We’ll preserve that numerator-denominator direction, reduce the quotient, interpret the within-A rate, …
Preview problemFind a joint probability by multiplying a marginal probability and a conditional probability
Apply and interpret the general Multiplication Rule with conditional probability.
A probability-tree path represents a sequence of branch conditions, so its probability comes from multiplication along that single path. We’ll trace A first and B after A, read the marginal …
Preview problemBuild a two-stage probability tree and calculate a named path
Apply and interpret the general Multiplication Rule with conditional probability.
A probability tree must attach each second-stage rate to the branch that supplies its condition. We’ll complete every complementary split to one, keep the two B rates on their matching …
Preview problemCompare matched independent and dependent joint probabilities
Apply and interpret the general Multiplication Rule with conditional probability.
Replacement determines whether the second draw resets or changes the deck. We’ll use the same first-red probability in both experiments, keep the second rate unchanged with replacement, update both counts …
Preview problemFind the probability of an ordered compound event using the multiplication rule
Apply and interpret the general Multiplication Rule with conditional probability.
An ordered without-replacement event uses a conditional second factor based on what left the collection first. We’ll form the red-first probability, update the remaining total while keeping every blue object, …
Preview problemFind a compound event probability by multiplying along each path and adding disjoint paths
Apply and interpret the general Multiplication Rule with conditional probability.
Exactly one head and one tail can occur along two different ordered paths. We’ll identify both successful routes, multiply along each fair-coin path, verify that the routes are disjoint, and …
Preview problemInterpret an unordered intersection or ordered joint path in context
Apply and interpret the general Multiplication Rule with conditional probability.
The word and identifies an intersection measured against the full population unless a conditioning phrase is present. We’ll name both required student properties, distinguish the joint event from a within-studied …
Preview problemCount permutations when order matters
Use permutations and combinations to compute probabilities of compound events and solve problems.
When all objects are distinct and every position matters, the available choices decrease one position at a time. We’ll multiply those counts, recognize the descending product as a factorial, evaluate …
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